Question

a# review a (topics 2.13 - 2.15) log equations and semi-log plots name: semi - log plot with y - axis from 1 to 1000 (1, 10, 100, 1000) and x - axis from 0 to 5 21. a set of data is shown on the semi - log plot above. if an exponential regression is used to model the data, which of the following claim and explanation statements best fit these data? (a) the residual plot will show a clear pattern, because the exponential model was appropriate. (b) the residual plot will show no clear pattern, because the exponential model was appropriate. (c) the residual plot will show a clear pattern, because the exponential model was not appropriate. (d) the residual plot will show no clear pattern, because the exponential model was not appropriate.

Explanation:

1. Recall the concept of residual plots and model appropriateness: A residual plot shows the difference between observed and predicted values. If a model (like exponential regression) is appropriate, the residuals should be randomly scattered (no clear pattern). If not, a pattern appears.
2. Analyze the semi - log plot: In a semi - log plot for exponential data (where \(y = ab^{x}\), and we take log of \(y\) to linearize it as \(\log y=\log a + x\log b\)), if the data points lie close to a straight line on the semi - log plot, the exponential model is appropriate.
3. Evaluate the options:

• Option A: If the model is appropriate, residuals should have no pattern. So A is wrong.
• Option B: If the exponential model is appropriate (data follows exponential trend, seen from semi - log plot linearity), residual plot has no clear pattern. This matches.
• Option C: If model is appropriate, no pattern. So C is wrong.
• Option D: If model is not appropriate, there would be a pattern, but here model is appropriate. So D is wrong.

Answer:

B. The residual plot will show no clear pattern, because the exponential model was appropriate.